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Question

In a plane there are 37 straight lines, of which 13 pass through the point A and 11 pass through the point B. Besides, no three lines pass through one point, no lines passes through both points A and B, and no two are parallel, then the number of intersection points the lines have is equal to

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Solution

The correct option is **A** 535

Let each line that passes through point A be known as an A line.

Let each line that passes through point B be known as a B line.

Let each line that passes through neither point A nor point B be known as an N line.

Since there 13 A lines, 11 B lines, and a total of 37 lines, the number of N lines =37−13−11=13

Case 1: An A line intersects with a B line

Number of options for the A line =13

Number of options for the B line =11

To combine these options, we multiply:

13×11=143

Case 2: An N line intersects with an A or B line

Number options for the N line =13

Number of options for the A or B line =13+11=24

To combine these options, we multiply:

13×24=312

Case 3: An N line intersects with another N line

Each PAIR of N lines will yield an intersection.

From the 13 N lines, the number of ways to choose2=13C2=(13×12)/(2×1)=78

Case 4: Points A and B

Points A and B constitute 2 more intersections =2

Total intersections =143+312+78+2=535

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